Nonlinear Control Design in the Frequency Domain.

A first step is made towards a complete generalization of the classical linear frequency domain theory of feedback control. First, the theory of partial fraction expansions is extended to multi-dimensional complex rational functions. As expected, the theory is now much more complicated and requires the use of ideal theory and notions from algebraic geometry. It turns out that, as in the linear case, the coefficients of the expansion (which are now polynomials) are obtained by "removing the given singularity" and evaluating on the single variety, i.e. evaluating the rational function modulo the singularity in the coordinate ring of the singularity. Multi-dimensional residue theory is based on the use of homology groups of the space ( in fact, the compactified version S2n) minus the singularity version T. We shall show that the inversion of the n-dimensional Laplace transform can be performed by finding a homology basis pf Hq (S2n\T), and a dual basis of Hr-1 (TU {oo}), where r+q=n. This will reduce the computation in many cases to a simple application of the n-dimensional version of Cauchy's theorem. The use of the theory in feedback control design is given with a particular study of a simple second-order bilinear system. We shall define an imlicit closed-loop transfer function (which is nonseparable) and then apply norm inequalities in the time domain to complete the stability analysis.